Contributions to the study of second order mean field games

The main theme of this thesis are Mean Field Games. We focus on two main aspects of the theory. The first one is the study of the so-called stable solutions. Our approach consists in rephrasing the problem as finding zeros of some nonlinear mapping defined on Banach spaces. The stability of solutions can then be expressed as the fact that the differential is an isomorphism. This approach allows us to prove the existence of finite element approximations, with error estimates, of the stable solutions. To achieve this, we rely on the Brezzi-Rappaz-Raviart approximation theorem, which we generalize to the case of nonsmooth problems by making use of the theory of metrically regular mappings. The second one has to do with the analysis of mean field games obtained when the players are modeled by sticky diffusion processes on networks. For this, we to study the properties of these stochastic processes and to prove an Itô formula.

PhD Thesis – INSA Rennes